MODEL PROBLEMS IN NUMERICAL STABILITY THEORY FOR INITIAL-VALUE PROBLEMS

In the past numerical stability theory for initial value problems in o rdinary differential equations has been dominated by the study of prob lems with simple dynamics; this has been motivated by the need to stud y error propagation mechanisms in stiff problems, a question modeled e ffectively by contractive linear or nonlinear problems. While this has resulted in a coherent and self-contained body of knowledge, it has n ever been entirely clear to what extent this theory is relevant for pr oblems exhibiting more complicated dynamics. Recently there have been a number of studies of numerical stability for wider classes of proble ms admitting more complicated dynamics. This on-going work is unified and, in particular, striking similarities between this new developing stability theory and the classical linear and nonlinear stability theo ries are emphasized. The classical theories of A, B and algebraic stab ility for Runge-Kutta methods are briefly reviewed; the dynamics of so lutions within the classes of equations to which these theories apply- linear decay and contractive problems-are studied. Four other categori es of equations-gradient, dissipative, conservative and Hamiltonian sy stems-are considered. Relationships and differences between the possib le dynamics in each category, which range from multiple competing equi libria to chaotic solutions, are highlighted. Runge-Kutta schemes that preserve the dynamical structure of the underlying problem are sought , and indications of a strong relationship between the developing stab ility theory for these new categories and the classical existing stabi lity theory for the older problems are given. Algebraic stability, in particular, is seen to play a central role. It should be emphasized th at in all cases the class of methods for which a coherent and complete numerical stability theory exists, given a structural assumption on t he initial value problem, is often considerably smaller than the class of methods found to be effective in practice. Nonetheless it is argua ble that it is valuable to develop such stability theories to provide a firm theoretical framework in which to interpret existing methods an d to formulate goals in the construction of new methods. Furthermore, there are indications that the theory of algebraic stability may somet imes be useful in the analysis of error control codes which are not st able in a fixed step implementation; this work is described.